The spectrum of quantum-group-invariant transfer matrices

Abstract

Integrable open quantum spin-chain transfer matrices constructed from trigonometric R-matrices associated to affine Lie algebrasĝ, and from certain K-matrices (reflection matrices) depending on a discrete parameter p, were recently considered in arXiv:1802.04864 and arXiv:1805.10144. It was shown there that these transfer matrices have quantum group symmetry corresponding to removing the p th node from theĝ Dynkin diagram. Here we determine the spectrum of these transfer matrices by using analytical Bethe ansatz, and we determine the dependence of the corresponding Bethe equations on p. We propose formulas for the Dynkin labels of the Bethe states in terms of the numbers of Bethe roots of each type.We also briefly study how duality transformations are implemented on the Bethe ansatz solutions.

Factorization identities and algebraic Bethe ansatz for $$ {D}_2^{(2)} $$ models

“…In particular, we construct the models' Bethe states, which had not been known, that would be needed to compute scalar products and correlation functions. Moreover, we prove previouslyproposed expressions for the models' eigenvalues and Bethe equations [16,[25][26][27][28]. The interesting degeneracies exhibited by these models are also explained.…”

Towards the solution of an integrable $D_{2}^{(2)}$ spin chain

“…either from the functional relation (3.7), or by explicitly computing the reference-state eigenvalue for small values of N and with values of the inhomogeneities chosen such that only Z 1 (u) is nonzero, as explained in detail in [11]. Note that a(u) (3.16) has a double-pole at u = η + iπ 2 .…”

“…These Bethe equations are unusual, as they involve cosh instead of sinh on the RHS. (For the ε = 0 case [7,9,11], the Bethe equations are the same as (5.5) except with sinh on the RHS. )…”

“…These spin chains also have a p ↔ n − p duality symmetry. Bethe ansatz solutions for the open D (2) n+1 spin chains with ε = 0 (and all the possible values of p) were proposed in [9,11]. However, the open D (2) n+1 spin chains with ε = 1 have so far resisted solution.…”

“…For the case ε = 0[9,11], and presumably also for the periodic chain [5], such a factorization is possible for all the eigenvalues, hence it may hold at the level of the transfer matrix.…”